Generalized method of moments estimation in Stata 11

Transcription

1 Generalized method of moments estimation in Stata 11 David M. Drukker StataCorp Stata Conference Washington, DC / 27

2 Outline 1 A quick introduction to GMM 2 gmm examples Ordinary least squares Two-stage least squares Cross-sectional Poisson with endogenous covariates Fixed-effects Poisson regression 2 / 27

3 A quick introduction to GMM Method of Moments (MM) We estimate the mean of a distribution by the sample mean, the variance by the sample variance, etc We want to estimate µ = E[y] We use µ = (1/N) N i=1 y i This estimator has nice properties because it solves the sample moment condition (1/N) N (y i µ) = 0 i=1 which is the sample analog of the population moment condition E[y µ] = 0 Estimators that solve sample moment equations to produce estimates are called method-of-moments (MM) estimators This method dates back to Pearson (1895) 3 / 27

4 A quick introduction to GMM Generalized method-of-moments (GMM) The MM only works when the number of moment conditions equals the number of parameters to estimate If there are more moment conditions than parameters, the system of equations is algebraically over identified and cannot be solved Generalized method-of-moments (GMM) estimators choose the estimates that minimize a quadratic form of the sample moment conditions GMM gets as close to solving the over-identified system of sample moment equations as possible GMM reduces to MM when the number of parameters equals the number of moment conditions Hansen (1982) produced many of the key results; Wooldridge (2002); Cameron and Trivedi (2005) provide good introductions 4 / 27

5 A quick introduction to GMM Definition of GMM estimator 5 / 27 Our research question implies q population moment conditions E[m(w i, θ)] = 0 m is q 1 vector of functions whose expected values are zero in the population w i is the data on person i θ is k 1 vector of parameters, k q The sample moments that correspond to the population moments are m(θ) = (1/N) N i=1 m(w i, θ) When k < q, GMM chooses the parameters that are as close as possible to solving the over-identified system of moment equations θ GMM arg min θ m(θ) Wm(θ)

6 A quick introduction to GMM Some properties of the GMM estimator θ GMM arg min θ m(θ) Wm(θ) When k = q, the MM estimator solves m(θ) exactly so m(θ) Wm(θ) = 0 W only affects the efficiency of the GMM estimator Setting W = I yields consistent, but inefficient estimates Setting W = Cov[m(θ)] 1 yields an efficient GMM estimator We can take multiple steps to get an efficient GMM estimator 1 Let W = I and get θ GMM1 arg min θ m(θ) m(θ) 2 Use θ GMM1 to get Ŵ, which is an estimate of Cov[m(θ)] 1 3 Get θ GMM2 arg min θ m(θ) Ŵm(θ) 4 Repeat steps 2 and 3 using θ GMM2 in place of θ GMM1 6 / 27

7 gmm examples The gmm command The new command gmm estimates parameters by GMM gmm is similar to nl, you specify the sample moment conditions as substitutable expressions Substitutable expressions enclose the model parameters in braces {} 7 / 27

8 gmm examples The interactive syntax of gmm Ordinary least squares For many models, the population moment conditions have the form E[ze(β)] = 0 where z is a q 1 vector of instrumental variables and e(β) is a scalar function of the data and the parameters β The corresponding syntax of gmm is gmm (eb expression) [ if ][ in ][ weight ], instruments(instrument varlist) [ options ] where some options are onestep use one-step estimator (default is two-step estimator) winitial(wmtype) initial weight-matrix W wmatrix(witype) weight-matrix W computation after first step vce(vcetype) vcetype may be robust, cluster, bootstrap, hac 8 / 27

9 gmm examples Ordinary least squares Ordinary least squares (OLS) is an MM estimator We know that OLS estimates the parameters of the conditional expectation of y i = x i β + ǫ i under the assumption that E[ǫ x] = 0 Standard probability theory implies that E[ǫ x] = 0 E[xǫ] = 0 So the population moment conditions for OLS are E[x(y xβ)] = 0 The corresponding sample moment conditions are Solving for β yields (1/N) N i=1 x i(y i x i β) = 0 β OLS = ( N i=1 x i x i ) 1 N i=1 x i y i 9 / 27

10 gmm examples Modeling crime data I Ordinary least squares We have (fictional) data on crime in 3,000 communities. use cscrime2, clear. describe Contains data from cscrime2.dta obs: 3,000 vars: 5 29 Jul :02 size: 132,000 (98.7% of memory free) (_dta has notes) storage display value variable name type format label variable label policepc double %10.0g police officers per thousand arrestp double %10.0g arrests/crimes convictp double %10.0g convictions/arrests legalwage double %10.0g legal wage index 0-20 scale crime double %10.0g property-crime index 0-50 scale Sorted by: 10 / 27

11 gmm examples Modeling crime data II Ordinary least squares We specify that crime i = policepc i β 1 +legalwage i β 2 + β 3 + ǫ i We want to model E[crime policepc,legalwage] = policepcβ 1 +legalwageβ 2 +β 3 If E[ǫ policepc,legalwage] = 0, the population moment conditions are [( ) ] policepc E (crime policepcβ legalwage 1 legalwageβ 2 β 3 ) = ( ) / 27

12 OLS by GMM I gmm examples Ordinary least squares. gmm (crime - policepc*{b1} - legalwage*{b2} - {b3}), /// > instruments(policepc legalwage) nolog Final GMM criterion Q(b) = 2.62e-31 GMM estimation Number of parameters = 3 Number of moments = 3 Initial weight matrix: Unadjusted Number of obs = 3000 GMM weight matrix: Robust Robust Coef. Std. Err. z P> z [95% Conf. Interval] /b /b /b Instruments for equation 1: policepc legalwage _cons 12 / 27

13 OLS by GMM II gmm examples Ordinary least squares. regress crime policepc legalwage, robust Linear regression Number of obs = 3000 F( 2, 2997) = Prob > F = R-squared = Root MSE = Robust crime Coef. Std. Err. t P> t [95% Conf. Interval] policepc legalwage _cons / 27

14 IV and 2SLS gmm examples Two-stage least squares For some variables, the assumption E[ǫ x] = 0 is too strong and we need to allow for E[ǫ x] 0 If we have q variables z for which E[ǫ z] = 0 and the correlation between z and x is sufficiently strong, we can estimate β from the population moment conditions E[z(y xβ)] = 0 z are known as instrumental variables If the number of variables in z and x is the same (q = k), solving the the sample moment conditions yields the MM estimator known as the instrumental variables (IV) estimator If there are more variables in z than in x (q > k) and we let ( N 1 W = i=1 z i i) z in our GMM estimator, we obtain the two-stage least-squares (2SLS) estimator 14 / 27

15 2SLS on crime data I gmm examples Two-stage least squares The assumption that E[ǫ policepc] = 0 is false if communities increase policepc in response an increase in crime (an increase in ǫ i ) The variables arrestp and convictp are valid instruments, if they measure some components of communities toughness-on crime that are unrelated to ǫ but are related to policepc We will continue to maintain that E[ǫ legalwage] = 0 15 / 27

16 2SLS by GMM I gmm examples Two-stage least squares. gmm (crime - policepc*{b1} - legalwage*{b2} - {b3}), /// > instruments(arrestp convictp legalwage ) nolog onestep Final GMM criterion Q(b) = GMM estimation Number of parameters = 3 Number of moments = 4 Initial weight matrix: Unadjusted Number of obs = 3000 Robust Coef. Std. Err. z P> z [95% Conf. Interval] /b /b /b Instruments for equation 1: arrestp convictp legalwage _cons 16 / 27

17 2SLS by GMM II gmm examples Two-stage least squares. ivregress 2sls crime legalwage (policepc = arrestp convictp), robust Instrumental variables (2SLS) regression Number of obs = 3000 Wald chi2(2) = Prob > chi2 = R-squared =. Root MSE = Robust crime Coef. Std. Err. z P> z [95% Conf. Interval] policepc legalwage _cons Instrumented: policepc Instruments: legalwage arrestp convictp 17 / 27

18 gmm examples Poisson with endogenous covariates Cross-sectional Poisson with endogenous covariates We want to model to E[y i x i, ν i ] = exp(x i β)ν i This setup allows the distribution of ν i to depend on x i Mullahy (1997) showed that we can use instrumental variables z i and the population moment conditions to estimate β E [z i (y i exp(x i β) 1)] = 0 18 / 27

19 gmm examples Cross-sectional Poisson with endogenous covariates. use accident2, clear. describe Contains data from accident2.dta obs: 948 vars: 6 29 Jul :59 size: 26,544 (99.7% of memory free) storage display value variable name type format label variable label kids cvalue tickets traffic male accidents float %9.0g float %9.0g float %9.0g float %9.0g float %9.0g float %9.0g Sorted by: traffic and male are exogenous variables tickets is an endogenous variable kids and cvalue are instrumental variables 19 / 27

20 gmm examples Cross-sectional Poisson with endogenous covariates. gmm (accidents*exp(-tickets*{b1} - traffic*{b2} - male*{b3} - {b4}) - 1), /// > instruments(kids cvalue traffic male) onestep nolog Final GMM criterion Q(b) = GMM estimation Number of parameters = 4 Number of moments = 5 Initial weight matrix: Unadjusted Number of obs = 948 Robust Coef. Std. Err. z P> z [95% Conf. Interval] /b /b /b /b Instruments for equation 1: kids cvalue traffic male _cons 20 / 27

21 gmm examples Fixed-effects Poisson regression More complicated moment conditions The structure of the moment conditions for some models is too complicated to fit into the interactive syntax used thus far For example, Wooldridge (1999, 2002); Blundell, Griffith, and Windmeijer (2002) discuss estimating the fixed-effects Poisson model for panel data by GMM. In the Poisson panel-data model we are modeling E[y it x it, η i ] = exp(x it β + η i ) Hausman, Hall, and Griliches (1984) derived a conditional log-likelihood function when the outcome is assumed to come from a Poisson distribution with mean exp(x it β + η i ) and η i is an observed component that is correlated with the x it 21 / 27

22 gmm examples Fixed-effects Poisson regression Wooldridge (1999) showed that you could estimate the parameters of this model by solving the sample moment equations ( ) i t x y it y it µ i it = 0 µ i These moment conditions do not fit into the interactive syntax because the term µ i depends on the parameters Need to use moment-evaluator program syntax 22 / 27

23 gmm examples Fixed-effects Poisson regression Moment-evaluator program syntax An abbreviated form of the program syntax for gmm is gmm moment program [ if ][ in ][ weight ], equations(moment cond names) parameters(parameter names) [ instruments() options ] The moment program is an ado-file of the form program gmm_eval version 11 syntax varlist if, at(name) quietly { <replace elements of varlist with error part of moment conditions> } end 23 / 27

24 Panel Accident data gmm examples Fixed-effects Poisson regression. use xtaccidents. describe Contains data from xtaccidents.dta obs: 5,000 vars: 7 31 May :50 size: 160,000 (98.5% of memory free) storage display value variable name type format label variable label id male t kids cvalue tickets accidents float %9.0g float %9.0g float %9.0g float %9.0g float %9.0g float %9.0g float %9.0g Sorted by: id t. by id: egen max_a = max(accidents ). drop if max_a ==0 (3750 observations deleted) 24 / 27

25 gmm examples Fixed-effects Poisson regression program xtfe version 11 syntax varlist if, at(name) quietly { tempvar mu mubar ybar generate double mu = exp(kids* at [1,1] /// + cvalue* at [1,2] /// + tickets* at [1,3]) if egen double mubar = mean( mu ) if, by(id) egen double ybar = mean(accidents) if, by(id) replace varlist = accidents /// - mu * ybar / mubar if } end 25 / 27

26 FE Poisson by gmm gmm examples Fixed-effects Poisson regression. gmm xtfe, equations(accidents) parameters(kids cvalue tickets) /// > instruments(kids cvalue tickets, noconstant) /// > vce(cluster id) onestep nolog Final GMM criterion Q(b) = 1.50e-16 GMM estimation Number of parameters = 3 Number of moments = 3 Initial weight matrix: Unadjusted Number of obs = 1250 (Std. Err. adjusted for 250 clusters in id) Robust Coef. Std. Err. z P> z [95% Conf. Interval] /kids /cvalue /tickets Instruments for equation 1: kids cvalue tickets 26 / 27

27 gmm examples Fixed-effects Poisson regression FE Poisson by xtpoisson, fe. xtpoisson accidents kids cvalue tickets, fe nolog Conditional fixed-effects Poisson regression Number of obs = 1250 Group variable: id Number of groups = 250 Obs per group: min = 5 avg = 5.0 max = 5 Wald chi2(3) = Log likelihood = Prob > chi2 = accidents Coef. Std. Err. z P> z [95% Conf. Interval] kids cvalue tickets / 27

28 References Bibliography Blundell, Richard, Rachel Griffith, and Frank Windmeijer Individual effects and dynamics in count data models, Journal of Econometrics, 108, Cameron, A. Colin and Pravin K. Trivedi Microeconometrics: Methods and applications, Cambridge: Cambridge University Press. Hansen, L. P Large-sample properties of Generalized Method-of-Moment Estimators, Econometrica, Hausman, Jerry A., Bronwyn H. Hall, and Zvi Griliches Econometric Models for Count Data with an Application to the Patents-R&D Relationship, Econometrica, 52(4), Mullahy, J Instrumental variable estimation of Poisson Regression models: Application to models of cigarette smoking behavior, Review of Economics and Statistics, 79, Pearson, Karl Contributions to the mathematical theory of evolution II. Skew variation in homogeneous material, 27 / 27

29 gmm examples Fixed-effects Poisson regression Philosophical Transactions of the Royal Society of London, Series A, 186, Wooldridge, Jeffrey Econometric Analysis of Cross Section and Panel Data, Cambridge, Massachusetts: MIT Press. Wooldridge, Jeffrey M Distribution-free estimation of some nonlinear panel-data models, Journal of Econometrics, 90, / 27